Hello Annie. Given the fact that you are given two conditions for this problem, we can assume that this problem is going to be solving a system of equations by either substitution or elimination.
For the problem we can assign x to be Ellie's age and y to be Genya's age. Given the first condition we can assign this to be the first equation: x + y = 60.
Using the second condition we have to do some manipulation. After 6 years, the women's ages increase by x + 6 and y + 6, and Genya's age is now eight times greater than Ellie's. So now we can rewrite the equation as such:
8(x + 6) = y + 6
8x + 48 = y + 6
8x + 42 = y
8x - y = -42
So we can now write the system of equations like this:
x + y = 60
8x - y = -42
In order to eliminate one of the variables you have to add them together with one of the variables being the opposite in sign to its counterpart in the other equation. In this instance you can see that the y in the top equation is positive and the y in the bottom equation is negative, so that step is taken care of. When you add them together, you get:
x + y = 60
+ 8x - y = -42
---------------------
9x = 18
After isolating x, you get: x = 2, or Ellie's age being 2.
And when you plug it in to your original condition x + y = 60, you get y = 58, or Ganye's age being 54.
You can check your answers by adding 6 to both of these numbers ( 2 + 6 = 8, 58 + 6 = 64) and seeing that 64 is indeed 8 times greater than 8.
I hope this helped and good luck!
